The subject of PT-symmetry and its areas of application have been blossoming
over the past decade. Here, we consider a nonlinear Schr\"odinger model with a
complex potential that can be tuned controllably away from being PT-symmetric,
as it might be the case in realistic applications. We utilize two parameters:
the first one breaks PT-symmetry but retains a proportionality between the
imaginary and the derivative of the real part of the potential; the second one,
detunes from this latter proportionality. It is shown that the departure of the
potential from the PT -symmetric form does not allow for the numerical
identification of exact stationary solutions. Nevertheless, it is of crucial
importance to consider the dynamical evolution of initial beam profiles. In
that light, we define a suitable notion of optimization and find that even for
non PT-symmetric cases, the beam dynamics, both in 1D and 2D -although prone to
weak growth or decay- suggests that the optimized profiles do not change
significantly under propagation for specific parameter regimes
